In this brief review, we shall focus mainly on the states that are sharply peaked along the classical trajectories, so that
the description of the “effective" dynamics of the universe becomes available [1]}, and the questions raised recently in [2]} are avoided. This includes the studies of
the “effective" dynamics of the homogeneous and isotropic mLQC-I and mLQC-II models, and their cosmological perturbations in the framework of the
dressed metric and hybrid approaches. We shall first clarify the issue regarding the ambiguities in the extension of the effective potential for the scalar perturbations across the quantum bounce,
and then pay particular attention to the differences among the three models, LQC, mLQC-I and mLQC-II,
and possible observational signals. It is important to note that initial conditions are another subtle and important issue not only in LQC but also in mLQCs. This includes two parts: (i) when to impose the initial conditions, and (ii)
which kind of initial conditions one can impose consistently. To clarify this issue, we discuss it at length by showing the (generalized) comoving Hubble radius in each model
and in each of the dressed and hybrid approaches. From this analysis, one can see clearly what initial conditions can and cannot be imposed at a chosen initial time.
